# orders and structure of classical groups

###### Theorem 1.
 $\begin{array}[]{cc}|SL(d,q)|=q^{\binom{d}{2}}\prod_{i=2}^{d}(q^{i}-1),&|PSL(d,% q)|=\frac{|SL(d,q)|}{(d,q-1)},\\ |GL(d,q)|=q^{\binom{d}{2}}\prod_{i=1}^{d}(q^{i}-1),&|PGL(d,q)|=\frac{|GL(d,q)|% }{q-1},\\ |\Gamma L(d,q)|=(q-1)q^{\binom{d}{2}}\prod_{i=1}^{d}(q^{i}-1),&|P\Gamma L(d,q)% |=\frac{|\Gamma L(d,q)|}{q-1}.\end{array}$

A proof of this theorem can proceed in many directions which each explain more of the structure of the these groups so each approach is considered equally valid and presented here. We list them from most elementary to most theoretical.

1. 1.

Elementary linear algebra proof (http://planetmath.org/ElementaryProofOfOrders)

2. 2.

Proof from subgroups of $GL(V)$ (http://planetmath.org/TheoryFromOrdersOfClassicalGroups2)

3. 3.

Proof from Lie theory/Chevally groups. [PENDING.]

Title orders and structure of classical groups OrdersAndStructureOfClassicalGroups 2013-03-22 15:56:45 2013-03-22 15:56:45 Algeboy (12884) Algeboy (12884) 10 Algeboy (12884) Topic msc 11E57 msc 05E15